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Quantum information processing, operational quantum logic, convexity,

and the foundations of physics

Los Alamos Technical Report LA-UR 03-1199

Howard Barnum ∗ a

aCCS-3: Modelling, Algorithms, and Informatics. Mail Stop B256Los Alamos National Laboratory, Los Alamos, NM 87545 USA. barnum@lanl.gov

Quantum information science is a source of task-related axioms whose consequences canbe explored in general settings encompassing quantum mechanics, classical theory, andmore. Quantum states are compendia of probabilities for the outcomes of possible oper-ations we may perform a system: “operational states.” I discuss general frameworks for“operational theories” (sets of possible operational states of a system), in which convexityplays key role. The main technical content of the paper is in a theorem that any suchtheory naturally gives rise to a “weak effect algebra” when outcomes having the sameprobability in all states are identified, and in the introduction of a notion of “operationalgebra” that also takes account of sequential and conditional operations. Such frame-works are appropriate for investigating what things look like from an “inside view,” i.e.for describing perspectival information that one subsystem of the world can have aboutanother. Understanding how such views can combine, and whether an overall “geomet-ric” picture (“outside view”) coordinating them all can be had, even if this picture is verydifferent in structure from the perspectives within it, is the key to whether we may beable to achieve a unified, “objective” physical view in which quantum mechanics is theappropriate description for certain perspectives, or whether quantum mechanics is trulytelling us we must go beyond this “geometric” conception of physics.

Keywords: quantum information ; foundations of quantum mechanics ; quantum com-putation ; quantum logic ; convexity ; operational theories PACS codes:

1. Introduction

The central question quantum mechanics raises for the foundations of physics is whetherthe attempt to get a physical picture, from “outside” the observer, of the observer’s

∗Thanks to the US DOE for financial support.

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interaction with the world, a picture which views the observer as part of a reality whichis at least roughly described by some mathematical structure, which is interpreted bypointing out where in this structure we, the observers and experimenters, show up, andwhy things end up looking as they do to observers in our position, is doomed. The“relative state” picture that arises when one tries to describe the whole shebang by anobjectively existing quantum state is unattractive, and many seek to interpret quantumstates instead as subjective, “information” about how our manipulations of the worldcould turn out. Whatever else they may be the quantum states of systems clearly arecompendia of probabilities for the outcomes of possible operations we may perform on thesystems: “operational states.” An operational theory is a specification of the set of possibleoperations on a system and a set of admissible operational states. This “operational”point of view can be useful whether one wants to consider the operational theory as forsome reason all we can hope for, or as a description of how perspectives look within anoverarching theory such as the relative state interpretation (RSI).

While it has not yet made a decisive contribution toward resolving this tension, by fo-cussing on the role of information held (through entanglement or correlation) or obtained(by measurement) by one system about another QIP concentrates one’s attention on thepractical importance of such measurements, and develops flexibility in moving betweenthe inside and outside views of such information-gathering processes. It thus providestools and concepts, as well as the ever-present awareness, likely to be useful in resolvingthis tension, if that is possible.

This paper is dedicated to the memories of two researchers in quantum foundations, whoI knew only through their collaborators and their work: Rob Clifton and Gottfried T.(“Freddy”) Ruttiman. They will continue to influence and inspire for the duration of theintellectual adventure of understanding, at the deepest level, our theories of the world.Their work is particularly relevant to the themes of this paper. Algebraic quantum fieldtheory is an example of integrating local perspectives (local ∗-algebras of observables) intoa coherent overall structure; Clifton made deep investigations into foundational issues inAQFT—for example, Clifton and Halvorson (2001) considers entanglement in this setting.He was also involved in one of the most spectacular successes to date of the project ofapplying quantum information-theoretic axioms to quantum foundations (Clifton et al.,2002). Ruttiman’s work involved, for example, linearization theorems for lattice-basedquantum logics (Ruttiman, 1993) which parallel and prefigure the ones discussed hereinfor convex effect-algebras, and investigation of the relation between the property latticeand face lattice of a state space (Ruttiman, 1981).

The paper is organized as follows. Section 2 considers some salient general implications ofQIS for foundational questions (irrespective of its contributions to this project). Section3 discusses the relative state and “subjective” views on the foundations of quantum me-chanics. Section 4 discusses whether and how the perspectives of different observers canbe combined, via tensor products and other constructions. Section 5 constructs “weakeffect algebras” from probability compendia via identification of probabilistically equiva-lent outcomes, reviews operational quantum logic, especially convex effect algebras, and

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introduces the notion of operation algebra which formalizes the notion of doing operationsin sequence, possibly conditioned on the results of previous operations. In Section 7, Ibriefly consider uses of the framework in applying QIP ideas to foundational questions.

A major part of empirical quantum logic is “deriving quantum mechanics.” The hopeis that if this can be done with axioms whose operational, information-processing, orinformation-theoretic meaning is clear, then one will have a particularly nice kind ofanswer to the question “Why quantum mechanics?” QI/QC provides a source of ax-ioms, with natural interpretations involving the possibility or impossibility of information-theoretic tasks. This is likely to contribute to whichevermode of resolution turns out to beright. Within the “geometric” or “objective overall picture” resolution, one might obtainthe answer: Why quantum mechanics? “Because it’s the sort of structure you’d expect fordescribing certain perspectives (of the sort beings like us wind up with) that occur “fromthe inside point of view” within an overarching picture of this [fill in the blank] sort.”The blank might be filled in with a specific overarching physical theory, or with fairlygeneral features. A similar answer might arise from the more “subjectivist” point of viewon quantum states. Why quantum mechanics? “Because it’s the sort of structure you’dexpect for describing the perspectives “from the inside point of view” within a reality ofthis sort, which reality is however not completely describable in physical terms, so thatthese perspectives are as good as physics ever gets.” Those who anticipate or hope fora physical picture, including relative state-ers, and those who think such an overarchingphysical picture unlikely to emerge, can nevertheless fruitfully pursue similar projects us-ing axiomatic arguments involving the notion of “operational theory” to derive quantummechanics, to understand, how it differs from or is similar to other conceivable theories,and the extent to which it does or does not follow from elementary conceptual require-ments (one way in which it could be “a law of thought”) or, in a more Kantian or perhaps“anthropic” way, from the possibility of rational beings like us (a different way in whichit could be “a law of thought”). Details might depend on one’s orientation: subjectivistsmight be more inclined to axioms stressing the formal analogies between density matricesand probability distributions, and between quantum “collapse” and Bayesian updating ofprobability distributions (Fuchs, 2001a). But since on the “overarching physical picturewith perspectives” view the probabilities are also tied to a “subjective,” perspectival el-ement, the Bayesian analogy is quite natural on this picture too. The close link between“empirical operational theories” and perspectival information that one subsystem of theworld can have about another, and the importance of tasks, of what can and cannot bedone from a given perspective, suggests that generalized information theory and infor-mation processing, of which QIS supplies a main example, will play a major role in thisproject.

2. QIP: The power of the peculiar

Virtually all of the main aspects of quantum mechanics exploited in QIP protocols havebeen understood for decades to be important peculiarities of quantum mechanics. Thenonlocal correlations allowed by entanglement are exploited by better-than-classical com-

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munication complexity protocols (Buhrman et al., 1997); the necessity of disturbancewhen information is gathered on a genuinely quantum ensemble (Fuchs and Peres, 1995;Barnum, 1998, 2001; Banaszek, 2001; Bennett et al., 1994; Barnum et al., 2001), closelyrelated to the “no-cloning theorem” (Wootters and Zurek, 1982) and no-broadcasting the-orem (Barnum et al., 1996; Lindblad, 1999), is the basis of quantum cryptography; theability to obtain information complementary to that available in the standard compu-tational basis is the heart of the historic series of algorithms due to Deutsch (1985),Deutsch and Jozsa (1992), Simon (1997), Bernstein and Vazirani (1997), and culminat-ing in Shor’s (1994; 1997) polynomial-time factoring algorithm. These peculiarities areno longer just curiosities, paradoxes, philosophers’ conundrums, they now have worldlypower.

A number of more specific and/or technical points on which QIP has contributed, orshows potential to contribute, something new to old debates can be identified. First, QIPprovides tools with which to analyze much more precisely and algorithmically questionsof what can and cannot be measured (Wigner, 1952; Araki and Yanase, 1960; Reck et al.,1994), or otherwised accomplished, either precisely or approximately, in quantum me-chanics. Some measurements are even uncomputable in essentially the same sense as aresome partial recursive functions in classical computer science. This raises the issue ofthe extent to which “operational” limitations, including basic and highly theoretical onessuch as computability, should be built into our basic formalisms, and what it means forthe interpretation of those formalisms and the “reality” of the objects they refer to, ifthey are not. Second, QIP techniques and concepts such as error-correction and activeand passive stabilization and control promise to allow a much more systematic approachthan previously to experiments and thought-experiments suggested by foundational inves-tigations. Third, QIP has demonstrated the power of taking the formal analogy betweenquantum density matrices and classical probability distributions seriously. Most thingsone does with probability distributions in classical information theory have (sometimesmultiple) natural quantum analogues when quantum states replace probability distribu-tions. Fourth, QIP provides a source of natural “operational” questions about whethercertain information-processing tasks can or cannot be performed, usable when consid-ering empirical theories more general than quantum mechanics. Also, QIP may be anatural source of examples of empirical theories. These arise when one considers at-tempts to perform quantum information processing with the restricted meansavailable insome proposed implementation of quantum computing. For example, QIP considerationsstimulated some of us (Barnum et al., 2002) to generalize the notion of “entanglement”to pairs of lie algebras and beyond that to pairs of ordered linear spaces.

3. Relative state vs. information interpretation of quantum mechanics

The central tension in interpreting quantum mechanics is between the idea that we arepart of a quantum world, made of quantum stuff interacting with quantum stuff, evolvingaccording to the Schrodinger equation, and the apparent fact that when we evolve soas to correlate our state with that of some other quantum system which is initially in

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a superposition, we get a single measurement outcome, with probabilities given by thesquared moduli of coefficients of the projections of the state onto subspaces in whichwe see a definite measurement outcome. The RSI reconciles these ideas by taking theview that the experience of obtaining a definite measurement result is how things appearfrom one point of view, our subspace of the world’s Hilbert space, and the full stateof the world is indeed a superposition. As I see it the correct way, on this view, toaccount for the appearance that there is a single measurement result, is the idea that theexperience of a conscious history is associated with definite measurement results, so thatconsciousness forks when a quantum measurement is made (Barnum, 1990). Just as thereis no consciousness whose experience is that of the spacetime region occupied by you, me,Halley’s comet, and the left half of Georges Sand, so, after a measurement has correlatedme with the the z-spin of an initially x-polarized photon, there is no consciousness whoseexperience is that of the full superposition (or, once these branches of me are decohered,of the corresponding mixture). Understanding why this happens as it does would appearto involve psychological/philosophical considerations about how minds are individuated.A more precise account must await a better scientific understanding of consciousness,though there are probably some useful things to be said by philosophers, psychologists,biologists, and decoherence theorists. It is deeply bound up with the problem of choosinga “preferred basis” in the relative state interpretation (i.e., the question, “relative towhat?”), and also with the problem of what tensor factorization of Hilbert space to choosein relativizing states, which appears in this light as the question of which subsystems of theuniverse support consciousness. The stability of phenomena and their relations enforcedby decoherence may underly the ability to support consciousness.

Despite sometimes conceding when pressed that they can’t show the RSI is inconsistent,its opponents also sometimes claim it is inconsistent for an observer to view him or herselfas described by quantum theory (Fuchs and Peres, 2000). I am not aware of a rigorousargument for this, though. Even an argument within a toy model would be valuable.But ven if it is shown that it would be inconsistent for an observer herself to have acomplete quantum-mechanical description of herself, the system she is measuring, andthe part of the universe that decoheres her “in the pointer basis,” that does not show thatsuch a description is itself inconsistent. Similar “bizarre self-referential logical paradoxes”Fuchs and Peres (2000) seem just as threatening (or not) for a classical description.

Some Bilodeau (1996) think that QM is telling us we must abandon the “geometric”conception of physics as giving us an “outside view” of reality. But I think that rather thanjust welcoming the ability to view quantum mechanics as only appropriate to describingan observer’s perspective on a system, revelling in the subjectivity of it all, the way itperhaps leaves room for mind, freewill, etc... as unanalyzed primitives, it is still promisingto try to get a grip on these matters “from an outside point of view.” An analogy mightbe special relativity. Here, an overarching picture was achieved by taking seriously thefact that position and time measurements are done via operations, from the perspectiveof particular observers. The heart of the theory is to coordinate those perspectives intoa global Minkowski space structure, explaining in the process certain aspects of the localoperational picture (like restrictions on the values of velocity measurements). I don’t

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think that we should yet give up on an attempt at such coordination in the quantumcase, perhaps celebrating the supposed fact that quantum mechanics has shown us thatit will be impossible to achieve under the aegis of physics.

An important point brought out by the attempt at a relative state interpretation of quan-tum mechanics is the need to bring in, in addition to Hilbert space, notions of preferredsubsystems (“experimenter” and “system” perhaps also the “rest of the world”) or pre-ferred orthogonal subspace decompositions (choice of “pointer basis” (Zurek, 1981)). Itseems unlikely, as Benjamin Schumacher likes to point out, that a Hilbert space, Hamil-tonian, and initial state, will single out preferred subspace decompositions in which dy-namics looks nontrivial, hence the RSI should involve aspects of physics beyond Hilbertspace. Schumacher also points out that a Hamiltonian evolution on a Hilbert space can bemade to look trivial by a time-dependent change of basis. If one takes the view that “theclassical world” is supposed to emerge from this structure (Hilbert space, Hamiltonian,and initial state), then perhaps such transformations are legitimate. On the other hand,they are not wholly trivial: if one specifies a Hamiltonian dynamics on a Hilbert space,one is implicitly specifying two groups of canonical isomorphisms between a continuum ofHilbert spaces, continuously parametrized by time. One of them says what we mean by“same Hilbert space at different times,” providing a framework with respect to which wecan then define a Hamiltonian evolution specified by the other one. If we could pick out aset of subspaces that are special with respect to this structure, that would be interesting.I have doubts that we can; I also like Schumacher’s criticism that this specification of“two connections on a fiber bundle instead of just one” seems mathematically unnatural.But I am not wholly convinced by Schumacher’s criticisms. I view the RSI less as a wayof getting the classical world emerge from Hilbert space, and more as a way of givinga realistic interpretation to Hilbert space structure in the presence of additional struc-ture such as preferred bases or subsystem decompositions that represent other aspectsof physics. Schumacher views his arguments as showing that one needs these additionalaspects of physics—”handles on Hilbert space”—to get a canonical identification of, say,bases from one time to the next (say the spin-up/down basis in a given reference frame).He interprets this as showing the appropriateness of Hilbert space descriptions for sub-systems where the special structure lies in relations to other systems (such as measuringappartus), and the inappropriateness of the Hilbert space structure for the description ofthe whole universe. There are plenty of such non-Hilbert space aspects of physics, involv-ing symmetries, spacetime structure. The need for renormalization and the difficultieswith quantum gravity suggest some difficulty in squaring quantum mechanics with someof these “geometrical,” “outside” aspects of physics. Perhaps the distasteful aspects ofthe quantum-mechanical outside view may vanish once such a squaring, with whateverflexing is necessary from both sides, is accomplished.

Bell showed that nonlocal hidden variables are the only non-conspiratorial way to real-istically model the statistics of quantum measurements. (Non-conspiratorial refers to aprohibition on explaining the statistics of quantum measurements by correlations betweenthe hidden variables and what we “choose” to measure.) But when we are contemplatingquantizing the spacetime metric or otherwise unifying gravity and quantum mechanics,

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perhaps it is not too farfetched to imagine that spacetime and causality will turn out tobe emergent from a theory describing a structure at a much deeper level....if this structurecontains things whose effects, at the emergent level of spacetime, can be interpreted asthose of “nonlocal hidden variables,” this should hardly surprise or dismay us.

My view toward the RSI with macroscopic superpositions is much like Einstein’s towardtaking quantum mechanics as a complete physical theory: I just don’t think the universeis like that. Schulman (1997) proposes to retain essentially a one-Hilbert space, state-vector evolving according to the Schrodinger equation, no-collapse version of quantummechanics, interpreted realistically, but to bring in cosmology and statistical mechanicsand argue that symmetric consideration of final conditions along with the usual initialconditions (that the universe was once much denser and hotter than it is now) rules outmacroscopic superpositions. There is a lot to do to make this persuasive. It is certainlyan ingenious and appealing idea. And if it does work, I am fairly happy to retain therest of the relative state metaphysics, now that I will not be committed to the disturbingexistence of forking Doppelgangers in subspaces of Hilbert space decohered from me.

4. The combination of perspectives

We should continue to investigate both the inside and outside views of quantum systems,and in interpretational matters to pursue a better understanding both of the possibility ofviewing quantum theory as about the dynamics of information-like, perhaps subjective,states, and of the possibility of viewing it as about the sorts of entanglement and correla-tion relations that can arise between systems. A prime example of a worthwhile programalong the former lines is the Caves-Fuchs-Schack Bayesian approach; a prime example ofa worthwhile program along the latter lines is understanding how the probabilities for col-lapse can be understood within the RSI Deutsch (1999); Wallace (2002), also as somethinglike a Bayesian process of “gaining more information about which branch of the wave-function we are in.” The similarity between these two programs is an example of how theoperational approach is relevant to both: investigate quantum mechanics’ properties as atheory of perspectives of subsystems on other systems, without prejudging whether or notthe perspectives will turn out to be coordinatable into an overarching picture—indeed,while trying to ferret out how this might happen or be shown to be inconsistent, and howthis possibility or impossibility may be reflected in the operational, perspective-boundstructures.

The Rovelli-Smolin “relational quantum mechanics” approach suggests ways in whichquantum mechanics could be good for describing things from the point of view of subsys-tems, but not appropriate for the entire universe, but in which nevertheless there existsa mathematical structure—something like a topological quantum field theory (TQFT) orspin foam—in which these local subsytem points of view are coordinated into an overallmathematical structure which, while its terms may be radically different from those weare used to, may still be viewed as in some sense “objective.” It is still far from clear thatthis can allow us to avoid the more grotesque aspects (proliferating macroscopic superpo-

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sitions viewed as objectively existing) and remaining conceptual issues (how to identifya preferred tensor factorization, and/or preferred bases, in which to identify “relativestates”) of the Everett interpretation.

In TQFT’s or spin networks and generalizations, the description appropriate to “perspec-tives” is still Hilbert spaces, but only in special cases do these combine as tensor products.If we view a manifold as divided into “system” and “observer” via a cobordism, then asthe “observer” gets small enough, while the “system” gets larger, we start getting, not theincrease in Hilbert-space size to describe the system that we might expect as the systemgets larger, but a decrease in Hilbert-space size whose heuristic interpretation might bethat the observer has gotten so small that it no longer has the possibility of measuringall the operators needed to describe the “large” Hilbert space one might have expected.The Hilbert space does not describe the “large” rest of the world; it describes the relationbetween a small observer and the larger rest of the world.

In these theories, we might see how the quantum description of certain perspectives couldarise as a limiting case of some more general type of perspective, which necessarily alsoarises in an overarching structure that includes quantum-mechanical perspectives in aphysically reasonable way. Or we might see how a non-tensor product law of combinationof subsystems—quantum or not—could be relevant in some situations. This is just thesort of thing that operational quantum logic aspires to investigate, and that might berelated to the ability to perform, or not, information-processing tasks.

5. Frameworks for empirical operational theories

In this section I will introduce frameworks I find particularly useful for thinking about em-pirical operational theories. David Foulis (1998) has provided a good review of the generalarea of mathematical descriptions of operational theories (which he calls “mathematicalmetascience”). That review stresses concepts similar to those I use here, notably that ofeffect algebras,” introduced under this name by Foulis and Bennett (Foulis and Bennett,1994), but also, as “weak orthoalgebras” in Greuling (1989), and independently, in anorder-theoretic formulation, as “difference posets” (D-posets, for short) by Kopka and Chovanec(1994). Longer and more technical introductions are available in Foulis (2000) and Wilce(2000).

5.1. Probabilistic equivalence

My preferred approach to operational theories starts from the compendia of probabilities,that are empirically found to be possible for the different results of different possible opera-tions on a system, and constructs various more abstract structures for representing aspectsof empirical theories—effect algebras, classical probability event-spaces, C∗-algebraic rep-resentations, spaces of density operators on Hilbert spaces, orthomodular lattices, orwhat have you—from these. With most such types of abstract structures, the possibilityof constructing them from phenomenological theories (sets of compendia of probabilities

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for measurement outcomes) will impose restrictions on these sets of compendia, and thenature of these restrictions constitutes the empirical significance of the statement thatour empirical theory has this abstract structure. This approach promises to systematizeour understanding of a wide range of empirical structures and their relationships, bothmathematically and in their empirical significance. The relationship to the probabilitiesof experimental outcomes has always been a critical part of understanding these struc-tures as empirical theories. The space of “states” on such structures is also often a crucialaid to understanding their abstract mathematical structure. This is of a piece with thesituation in many categories of mathematical objects. [0, 1] is a particularly simple ex-ample of many categories of “empirical structure,” and a state is a morphism onto it;understanding the structure of some more complex object in the category in terms of theset of all morphisms onto this simple object is similar to, say, understanding the structureof a group in terms of its characters (morphisms to a particularly simple group).

In this project I make use of an idea which has come in for a fair amount of criticism, buthas been with us from early in the game (cf. e.g. (Mackey, 1963), Cooke and Hilgevoord(1981) (who even ascribe it to Bohr), Ludwig (1983a), Mielnik (1969) p. 14). This is thenotion of “probabilistic equivalence”: two outcomes, of different operational procedures,are viewed as equivalent, if they have the same probability “no matter how the system isprepared,” i.e., in all admissible states of the phenomenological theory. An interpretationof equivalent outcomes as “exhibiting the same effect of system on apparatus” is probablydue to Ludwig, perhaps motivating his term “effect” for these equivalence classes (atleast in the quantum case). It helps forestall the objection that two outcomes equivalentin this sense may lead to different probabilities (conditional on the outcomes) for theresults of further measurements. They are equivalent only as concerns the effect of thesystem on the apparatus and observer, not vice versa. The criticism implicitly supposesa framework in which operations may be performed one after the other, so that outcomesof such a sequence of N measurements are strings of outcomes a1a2...aN of individualmeasurements. Then a stricter notion of probabilistic equivalence may be introduced,according to which two outcomes x and y are equivalent if for every outcome a, b theprobability of axb is the same as that of ayb, in every state.

Before considering in detail the derivation of the structure of the set of probabilistic equiv-alence classes (“effects”) of an operational theory, I will introduce some of the abstractstructures we will end up with: effect algebras and “weak effect algebras,” motivatingthem (in the case of effect algebras) with classical and quantum examples.

Definition 1 An effect algebra is an object 〈E , 1,⊕〉, where E is a set of “effects,” 1 ∈ E ,and ⊕ is a partial binary operation on E which is (EA1) strongly commutative and (EA2)strongly associative. The qualifier “strongly,” which is not redundant only because ⊕ ispartial, indicates that if the sums on one side of the equations for commutativity andassociativity exist, so do those on the other side, and they are equal. In addition, (EA3)∀e ∈ E , ∃!f ∈ E (e⊕ f = u). (The exclamation point indicates uniqueness. We give thisunique f the name e′; it is also called the orthosupplement of e.) (EA4) a⊕ 1 is definedonly for a = 1′. (We will often call 1′ by the name “0”.)

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If we only require that the equalities specifying associativity (a⊕ (b⊕c) = (a⊕b)⊕c) andcommutativity (a⊕ b = b⊕ a) hold when both sides are defined, allowing the possibilitythat one is defined while the other is not, we call these “weak commutativity” and “weakassociativity.”

In the effect algebra E(H) of quantum mechanics (on a finite-dimensional Hilbert spaceH, say), E is the unit interval of operators e such that 0 ≤ e ≤ I on the Hilbert space,⊕ is ordinary addition of operators restricted to this interval (thus e ⊕ f is undefinedwhen e + f > I), 1 is the identity operator I, and e′ = I − e, so 0 is the zero operator.A classical example is the set F of “fuzzy sets” on a finite set Λ = λ1, ..., λd (whichare functions from Λ to [0, 1]), with ⊕ as ordinary pointwise addition of functions (i.e.defining f + g by (f + g)(x) = f(x) + g(x) except that f ⊕ g is undefined when f + g’srange is not contained in [0, 1]), and 1 the constant function whose value is 1. 〈F , 1,⊕〉is an effect algebra obviously isomorphic to the restriction of the quantum effect algebraon a d-dimensional Hilbert space to effects which are all diagonalizable in the same basis.These “fuzzy sets” may be interpreted as the outcomes of “fuzzy measurements” in asituation where there are d underlying potential atomic “sharp” measurement results or“finegrained outcomes,” but our apparatus may have arbitrarily many possible meterreadings, connected to these “atomic outcomes” by a noisy channel (stochastic matrixof transition probabilities, which are in fact the d values taken by the function (effect)representing a (not necessarily atomic) “outcome”.).

We consider various modifications of the effect algebra notion. We introduce “weak effectalgebras” which are EA’s in which strong associativity (EA2) is replaced by weak associa-tivity. An orthoalgebra instead adds the axiom (OA5) that x⊕x exists only for x = 0. Theprojectors on a quantum-mechanical system, with the same definitions of 1,⊕ as applyto more general POVM elements, are an example (as well as being a sub-effect algebra ofE(H)). Wilce considered “partial abelian semigroups,” (PASes) which require only (EA1)and (EA2); various combinations of additional requirements then give a remarkably widevariety of algebraic structures that have been considered in operational quantum logic,including effect algebras, test spaces, E-test spaces, and other things. In particular, aneffect algebra is a positive, unital, cancellative, PAS (see below).

A state ω on a weak effect algebra 〈E ,⊕, 1〉 is a function from E to [0, 1] satisfying:ω(a⊕ b) = ω(a)+ω(b) , ω(1) = 1 . A finite resolution of unity in a weak effect algebra (tobe interpreted as the abstract analogue of a measurement) is a set R such that ⊕a∈Ra = 1.So for a resolution of unity R,

∑a∈R ω(a) = 1: the probabilities of measurement results

add to one. A morphism from one WEA E to another F is a function φ : E → F suchthat φ(a⊕ b) = φ(a)⊕ φ(b); it is called faithful if in addition, φ(1E) = 1F , where 1E and1F are the units of E and F . [0, 1], with ⊕ addition restricted to the interval, is an effectalgebra, so a state on E is a faithful morphism from E .

I will attempt to avoid issues involving effect algebras and WEA’s where E is infiniteand infinite resolutions of unity are defined, though finite dimensional quantum mechan-ics is properly done that way. ((Feldman and Wilce, 1993), Bugajski et al. (2000) and

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Gudder and Greechie (2000), for example treat these issues.) To this end I will assumethat EA’s and WEA’s are locally finite: resolutions of unity in them have finite cardinal-ity. For finite d-dimensional quantum mechanics, most things should work the same if werestrict ourselves to work with resolutions of unity into d2 elements.

Now, I will relate this abstract structure to phenomenological theories, by showing thatone can derive a natural weak effect algebra from any phenomenological theory. Theoperation ⊕ of the weak effect algebra will be the image, under our construction, of thebinary relations OR (∨) in the standard propositional logics (one for each measurement)of propositions about the outcomes of a given measurement. (This is one justification forcalling effect algebras “logics”.)

In order to describe this construction, we first review Boolean algebras. A Boolean algebrais an orthocomplemented distributive lattice. A lattice is a structure 〈L,∨,∧〉, where Lis a set, ∨,∧ total binary operations on L with the following properties. Both operationsare associative, commutative, and idempotent (idempotent means, e.g., (a ∧ a = a)). Inaddition, together they are absorptive: a∧ (a∨ b) = a , a∨ (a∧ b) = a. ∨ is usually calledjoin, ∧ is usually called meet. These properties are satisfied by letting L be any powerset(the set of subsets of a given set), and the operations ∨,∧ correspond to ∪,∩. For L = 2X

(the power set of X) we call this lattice the subset lattice of X . An important alternativecharacterization of a lattice is as a set partially ordered by a relation we will call ≤. Ifevery pair (x, y) of elements have both a greatest lower bound (inf) and a least upperbound (sup) according to this ordering, we call these x∧y and x∨y, respectively, and theset is a lattice with respect to these operations. Also, for any lattice as defined above, wemay define a partial ordering ≤ such that ∧, ∨ are inf, sup, respectively, in the ordering.So the two characterizations are equivalent.

A lattice is said to be distributive if meet distributes over join: a∨(b∧c) = (a∨b)∧(a∨c) .(This statement is equivalent to its dual (the statement with ∧ ↔ ∨).) If L contains topand bottom elements with respect to ≤, we call them 1 and 0. They may be equivalentlybe defined via a = a ∧ 1, a = a ∨ 0 for all a ∈ L. We define b to be a complement of a ifa ∧ b = 0 and a ∨ b = 1. Complements are unique in distributive lattices, not necessarilyso in more general lattices. When all complements are unique, we write complementationas a unary relation (operation) ′; this relation is not necessarily total even in distributivelattices with 0, 1. A Boolean lattice, or Boolean algebra, is a distributive lattice with0, 1, in which every element has a complement. Any subset lattice L = 2X is a Booleanalgebra, with 0 = ∅ and 1 = X .

Definition 2 A (locally finite) phenomenological theory P is a set M of disjoint finitesets M , together with a set Ω of functions (“states”) ω from (all of) ∪M∈PM to [0, 1]such that for any M ,

∑x∈M ω(x) = 1.

M are the possible measurements; taking them to be disjoint means we are not allowingany a priori identification of outcomes of different measurement procedures. Ω is the setof phenomenologically admissible compendia of probabilities for measurement outcomes.

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The set M is an example of what Foulis calls a “test space”: a set T of sets T , where Tmay be interpreted as operations, (tests, procedures, whatever you want to call them) andthe elements t ∈ T as outcomes of these operations. (Without the interpretation, these arebetter known in mathematics as hypergraphs or set systems.) Call the set of all outcomesΛ := ∪T . In general test spaces the T need not be disjoint; here they are. Foulis callssuch test spaces “semiclassical.” (Sometimes a weak requirement of irredundancy, thatnone of these sets is a proper subset of another, is imposed on test spaces; it is automatichere.) States on test spaces are functions ω : Λ → [0, 1] such that

∑t∈T ω(t) = 1 for any

T . It is only when a phenomenological theory is defined as a set of states on a generalT , where a given outcome may occur in different measurements, that the question ofcontextuality (does the probability of an outcome depend on the measurement it occursin?) arises at the phenomenological level. By not admitting such a primitive notion of“same outcome,” but distinguishing outcomes according to the measurements they occurin, the construction we make will guarantee noncontextuality of probabilities even at thelater stage where the theory is represented by a more abstract structure in which theelements (effects, or operations) that play the role of outcomes may occur in differentoperations. Though the rest of our discussion ignores it, the question of whether therecan be convincing reasons for admitting a primitive notion of “same outcome” (basedperhaps on some existing theory in terms of which the operations and experiments of our“phenomenological theory” are described) is worth further thought. A related point isthat test spaces provide a framework in which we can implement a primitive notion of twooutcomes of different measurements being the same, but we cannot implement a notionof two outcomes of the same measurement being the same (up to, say, arbitrary labeling).A formalism in which one can is that of E-test spaces (the E is for effect). These aresets, not of sets of outcomes, but of multisets of outcomes. Multisets are just sets withmultiplicity: each element of the universe is not just in or out of the set, but in the set witha certain nonnegative integer multiplicity. Where sets can be described by functions fromthe universe to 0, 1 (their characteristic functions), multisets are described by functionsfrom U to N. The set of resolutions of unity in an effect algebra, shorn of its algebraicstructure, is an E-test space (whence the name). Not all E-test spaces are such that aneffect algebra can be defined on them; those that are are called algebraic. Sufficientlynice E-test spaces are prealgebraic, and can be completed to be algebraic by adding moremultisets without enlarging the universe (underlying set of outcomes).

To each phenomenological theory we may associate a set of Boolean algebras, one foreach measurement. We will call this set of Boolean algebras the “phenomenological logic”of the theory; note, though, that it is independent of the state-set Ω. These are justthe subset lattices of the sets M , or what I previously called the “propositional logics” ofstatements about the results of the measurements. We will distinguish them by subscriptson the connectives saying which measurement is referred to, e.g. ∧M (although this isredundant due to the disjointness of the measurements).

The phenomenological states ω of P naturally induce states (which we will also call ω)on the logic of P, via ω(a) = ω(a), ω(X) =

∑x∈X ω(x). They will satisfy ω(M) = 1

for each M , and ω(∅) = 0. We have, for example (x and y are now subsets of outcomes),

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ω(x∨My) = ω(x)+ω(y)−ω(x∧My), (which is equivalent to its dual). We call the elementsof the Boolean algebras of a phenomenological logic events, and we will refer to the set ofevents of P as V.

Definition 3 Events e, f are probabilistically equivalent, e ∼ f in a phenomenologicaltheory if they have the same probability under all states: ∀ω ∈ Ω, ω(e) = ω(f) .

∼ is obviously an equivalence relation (symmetric, transitive, and reflexive). Hence wecan divide it out of the set V, obtaining a set V/ ∼ =: E(P) of equivalence classes ofevents which we will call the effects of the theory P. (We have dependence on P, ratherthan just M, because although V depends on M but not Ω, ∼ depends also on Ω. ) Callthe canonical map that takes each element a ∈ V to its equivalence class, “e.” The imagese(M) of the measurements M under e are “measurements of effects.” Together they forman E-test space as defined above (a set of multisets). We now define on this space another“logic” which is, at least as far as possible, the simultaneous “image” under the map e ofeach of the Boolean algebras M . To this end, we introduce a binary operation ⊕ on theeffect space.

Definition 4 e1 ⊕ e2 := e(a ∨M b) for some a such that e1 = e(a), b such that e2 = e(b),and M such that a, b ∈ M but a ∩ b = ∅.

If no such a, b,M exist, ⊕ is undefined on the effect space. (If they do exist, we will saythey witness the existence of e1⊕e2.) As part of the proof of Theorem 1 we will show fromthe definition of the map e via probabilistic equivalence and the behavior of probabilitieswith respect to ∨M , that this definition is independent of the choice of a, b,M .

Let ωe denote the function from the effects to [0, 1] induced in the obvious way by a stateω on the Boolean algebra: effects being equivalence classes of things having the samevalue of ω, we let ωe take each equivalence class to ω’s value on anything in it.

Definition 5 A set of states Ω on a WEA E is separating if for x, y ∈ E , x 6= y ⇒ ∃ω ∈Ω(ω(x) 6= ω(y)).

Theorem 1 The set E(P) of effects of a phenomenological theory P with state-set Ω,equipped with the operation ⊕ of Def. 4 and the definition 1 = e(1M) (for some M)constitutes a weak effect algebra. There exist phenomenological theories for which this isproperly weak, i.e. not an effect algebra. For all ω ∈ Ω the functions ωe defined above arestates on the resulting weak effect algebra. Ωe := ωe|ω ∈ Ω is separating on E(P).

The proof is a straightforward verification of the axioms and the statements about statesfrom the definition, and an example for the second sentence.

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Proof: We begin by demonstrating ⊕ is in fact a partial binary operation. Thisis done by verifying the independence, asserted above, of the definition of ⊕ from thechoice of a, b,M and of 1 from M . Suppose e1 = e(a) = e(c), e2 = e(b) = e(d), a, b ∈M, c, d ∈ N, a 6= b, c 6= d, a ∧M b = 0, c ∧N d = 0. Consider any state ω on the set ofBoolean algebras which is also in Ω, the states of our phenomenological theory. By thedefinition of e, ω(a) = ω(c) and ω(b) = ω(d) ; therefore ω(a)+ω(b) = ω(c)+ω(d). Nowω(a ∨M b) = ω(a) + ω(b) because a ∨M b = 0, and similarly ω(c ∨N d) = ω(c) + ω(d).In other words, for any state ω ∈ Ω, ω(a ∨M b) = ω(c ∨N d), so a ∨M b and c ∨N d areprobabilistically equivalent, and correspond to the same effect.

Each Boolean algebra contains a distinguished element 1; by the definition of state on P,these have probability zero, and one, respectively, in all states. Hence they each map toa single effect, and these effects we will call 0 and u in the effect algebra (verifying laterthat 0 = 1′ in the weak effect algebra, so that it is consistent with the usual definition of0 in a WEA). Of course, ωe(1) = 1. It is also easy to see that ωe(x⊕ y) = ωe(x)⊕ ωe(y).Hence the ωe are states, as claimed. The set Ωe is obviously separating. To be pedantic,suppose there exist effects x, y having ωe(x) = ωe(y) for all ωe ∈ Ωe. By the definition ofωe, ωe(x) is the common value of ω on all e-preimages of x, and ωe(y) is the common valueof ω on all e-preimages of y. If these values are the same for all ωe, then the preimages ofx and of y are all in the same equivalence class, so x = y. Hence, Ωe is separating.

We now verify that ⊕ satisfies the weak effect algebra axioms.

(EA1) Strong commutativity: If a, b ∈M witness the existence of x⊕y as described in thedefinition of ⊕, by symmetry of ∨M and ∧M (which enter symmetrically in the definitionof ⊕) they also witness the existence of y ⊕ x and its equality with x⊕ y.

(WEA2) Weak associativity. Let a, b ∈ M, e(a) = x, e(b) = y, a ∩ b = ∅, so that a, bwitness the existence of x⊕ y, and also let c, d ∈ N and disjoint, e(c) = z, e(d) = x ⊕ y,so c, d witness the existence of (x⊕ y)⊕ z. Similarly let b′, c′ ∈ P witness the existence ofy⊕z and a′, f ∈ Q witness the existence of x⊕(y⊕z), so that e(a′) = x, e(f) = y⊕z, anda′, f are disjoint. Then ωe(x⊕y) = ω(a)⊕ω(b) and ωe((x⊕y))⊕z) = ω(a)+ω(b)+ω(c) .Also ωe(y⊕ z) = ω(b′)⊕ω(c′) = ω(b)⊕ω(c), so ωe((x⊕ (y⊕ z)) = ω(a′)⊕ω(f) = ω(a)⊕ω(b)⊕ω(c) . But ωe((x⊕y)⊕z) = ωe(x⊕(y⊕z)) for all ωe implies (x⊕y)⊕z = x⊕(y⊕z)by the fact that Ωe is separating.

(EA3) Define e′ to be e(a′), for any a such that e(a) = e, and a′ is a’s unique complementin the Boolean algebra of the measurement M containing it. Since for any state, ω(a′) =1 − ω(a) and this probability is independent of a as long as e(a) = e, e′ as thus definedis independent of which a is chosen. Moreover, since a ∧M a′ = 0 e ⊕ e′ ≡ e(a) ⊕ e(a′)is defined and equal to e(a ∨M a′) = e(1M) = 1, so that ′ as we just defined it satisfies(EA3).

(EA4) Note that x⊕1 is equal to e(a∨M 1M), for some M containing a and with unit 1M ,where a ∧M 1 = 0 and e(a) = x. But each M has a unique a such that a ∧M 1M = 0M ,namely 0M . So an x such that x⊕ 1 exists; it must be e(0M) = 0.

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This proves the first part of the theorem. We remark that 1′ ≡ e(1′) = e(0M), so defining0 as e(0M) for any M coincides with the usual effect algebra definition as 1′. We nowconstruct the counterexample required by the second part.

Consider a phenomenological theory consisting of states on the two atomic Boolean alge-bras:

M : ( a b ) ( f )N : ( c ) ( d ) ( g )

(1)

with the indicated a, ..., g being atoms of the Boolean algebras involved (“elementarymeasurement outcomes”). The vertical lining-up of parentheses in (1) visually indicatesconditions we will impose on the theory: that all states of our phenomenological theoryrespect ω(a ∨M b) = ω(c) and ω(f) = ω(d ∨N g); further, let our theory contain stateswith nonzero probability for each of a, b, c, d, f, g. There are plenty of perfectly goodempirical theories satisfying these constraints, but ⊕ on the effect set of such a theorywill not exhibit strong associativity: although e(a)⊕ e(b) exists and is equal to e(c), ande(c) ⊕ e(d) exists and is therefore equal to (e(a) ⊕ e(b)) ⊕ e(d), no effect h exists withe(h) = e(b)⊕ e(d).

Conjecture 1 (Completion conjecture for WEA’s) Let E be a WEA obtained froma phenomenological theory. A unique effect algebra E , which we call the completion ofE , can be constructed from E as follows. Whenever only one side of the associativityequation exists, impose the equation (extend ⊕ to contain the pair that would appear onthe other side). This can also be characterized as the smallest effect algebra containing Eas a sub-weak-effect-algebra (with the latter concept appropriately defined).

Thus the well-developed and attractive theory of effect algebras could be useful in thismore general context. The adjunction of these new relations and the new resolutions ofunity whose existence they imply is an interesting theoretical move. In constructing the-ories, we often suppose the existence of things that do not, at least initially, correspondto things in the available phenomenology. The idea of including all Hermitian operatorsas observables in quantum mechanics is an example; there has been much discussion ofwhether they are all operationally observable. This has motivated the search, often suc-cessful, for methods of measuring observables that had previously not been measured, andthe development of a general theory of algorithmic procedures for measurement. The con-jecture above might motivate the search for empirical methods of making measurementswhich would correspond to the additional resolutions of unity needed to make the initialWEA into an effect algebra. In any case, it is worth studying the nature of informationprocessing and information theory (if the latter still makes sense) in properly weak effectalgebras versus their completions.

We are now ready for a few remarks on the significance of Gleason’s theorem (Gleason,1957) in this context. Gleason’s theorem says that in Hilbert space dimension greater than

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two, if mutually exclusive quantum measurement results are associated with mutuallyorthogonal subspaces of a Hilbert space, and exhaustive sets of such measurements todirect sum decompositions of the space into such subspaces, and if the probability ofgetting the result associated to a given subspace in a given measurement is independentof the measurement in which it occurs (“noncontextual”) then the probabilities must begiven by the trace of the product of the projector onto the given subspace with a densityoperator. A similar theorem resolutions of unity into orthogonal projectors replaced byresolutions into arbitrary positive operators has been obtained by Busch (1999), andindependently by Caves, Fuchs, Mannes, and Renes (Fuchs, 2001a,b). In the next sectionwe will see how this theorem is a case of a general fact about convex effect algebras.

Sometimes Gleason-type theorems are used to justify the quantum probability law. Thenone must justify the assumptions that probabilities are noncontextual, and that they areassociated with orthogonal decompositions, or positive resolutions of unity, on a Hilbertspace. Although Theorem 1 gives structures (WEAs) much more general than Hilbertspace effect algebras (or their subalgebras consisting of projectors), it automatically re-sults in noncontextual probability laws. But he construction of WEAs in Theorem 1 startsfrom probabilities, so it would be circular to use it to justify noncontextuality in an appealto Gleason’s theorem to establish quantum probabilities. Rather, Theorem 1 says that wecan elegantly, conveniently represent any empirical theory by a set of noncontextual prob-ability assignments on a certain WEA (and, if the completion conjecture is correct, embedthis in an effect algebra). In the case of quantum theory, this general recipe provides boththe Hilbert space structure and the trace rule for probabilities, as a representation of thecompendium of “empirical” probabilities (perhaps somewhat idealized by the assumptionthat any resolution of unity can be measured) of quantum theory.

The generalization of Gleason-like theorems to weak effect algebras, effect algebras, andsimilar structures are theorems characterizing the full set of possible states on a givensuch structure, or class of such structures. In the particular case of a Hilbert space ef-fect algebra, the import of the B/CFMR theorem, from our operational point of view,is that the quantum states constitute the full state space of the “empirically derived”effect algebra. This is especially interesting since in other respects, the category of effectalgebras probably does not have enough structure to capture everything we would like itto about quantum mechanics: for example, the natural category-theoretic notion of ten-sor product of effect algebras (Dvurecenskij (1995); see also Wilce (1994, 1998)), appliedto effect algebras of finite dimensional Hilbert spaces, does not give the effect algebra ofthe tensor product Hilbert space (or of any Hilbert space), as one sees from a result inFuchs (2001a) (a similar result involving projectors only is in Foulis and Randall (1981)).Possibly relatedly, a natural category of morphisms for convex effect algebras, those in-duced by positive (order-preserving) linear maps on the underlying ordered linear space(see below), is larger in the quantum case than the “completely positive” maps usuallyconsidered reasonable for quantum dynamics. Nevertheless for a given Hilbert space effectalgebra, its set of all possible states is precisely the set of quantum states.

The role of Gleason-like results depends to some extent on point of view. In the project of

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exploiting the analogies between quantum states and Bayesian probabilities, they can playa nice conceptual role. Probabilities are, roughly, “the right way” (nonarchimedeanity is-sues aside) to represent uncertainty, and to represent rational preferences over uncertainclassical alternatives. In this “Bayesian” project, it would be very desirable to see quan-tum states as “the right way” to deal with uncertainty in a nonclassical situation: theHilbert space structure perhaps sums up the “nonclassicality of the situation,” and theprobabilities can be seen as just the consequence of “rationality” in that situation. Thissuggests that the “structure of the nonclassical situation” mentioned above might bedescribed in terms of measurement outcomes (sometimes called “propositions” or “prop-erties”) having probability zero or one; then Gleason’s theorem or analogues for other“property” structures, might give the set of possible probability assignments for such astructure. This is related to the “Geneva” approach to empirical theories (rooted in thework of Jauch and Piron on “property lattices”).

5.2. Convex effect algebras

It is natural to take the space of operations one may perform as convex. This representsthe idea that given any operationsM1 andM2, we can perform the operation (λ1M1, λ2M2)(where λi ≥ 0, λ1+λ2 = 1) in which we perform one ofM1 orM2, conditional on the out-come of flipping a suitably weighted coin (or, in more Bayesian terms, arrange to believethat these will be performed conditional on mutually exclusive events, to which we assignprobabilities λ1, λ2, that we believe to be independent of the results of measurements onthe system under investigation). If we looked at the coin face and saw the index “i” andobtained the outcome a ofMi, this should correspond to an outcome λa of (λ1M1, λ2M2),and any state should satisfy ω(λa) = λω(a).

Similar assumptions may be made at the level of effect algebra. For effect algebras con-structed via probabilistic equivalence, they will be consequences of the convexity as-sumptions on the initial phenomenological theory; this will be worked out elsewhere.One could also pursue the consequences of imposing a generalized convexity based ona more refined notion of “vector probabilities”, or other representations of uncertaintyby nonarchimedean order structures. Such generalized probabilities and utilities canresult from Savage-like representation theorems for preferences satisfying “rationality”axioms but not certain technical axioms that make possible real-valued representations(LaValle and Fishburn, 1992, 1996; Fishburn and LaValle, 1998). We will avoid such com-plications, but knowing about them may clarify the role of some technical conditions inresults to be discussed below.

Definition 6 A convex effect algebra is an effect algebra 〈E, u,⊕〉 with the additionalassumptions that for every a ∈ E and α ∈ [0, 1] ⊂ R there exists an element of E, call itαa, such that (C1) α(βa) = (αβ)a, (C2) If α+ β ≤ 1 then αa⊕ βa exists and is equal to(α + β)a, (C3) α(a⊕ b) = αa⊕ αb (again, the latter exists), (C4) 1a = a. The mappinga 7→ αa from [0, 1]× E to E is called the convex structure of the convex effect algebra.

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Gudder and Pulmannova (1998) showed that “any convex effect algebra admits a repre-sentation as an initial interval of an ordered linear space,” and in addition if the set ofstates on the algebra is separating, the interval is generating. To understand this result,we review the mathematical notion of a “regular” positive cone (which we will just callcone); it is basic in quantum information science, e.g because the quantum states, theseparable states of a multipartite quantum system, the completely positive maps, thepositive maps, unnormalized in each case, form such cones.

Definition 7 A positive cone is a subset K of a real vector space V closed under multipli-cation by positive scalars. It is called regular if it is (a) convex (equivalently, closed underaddition: K + K = K), (b) generating (K − K = V , equivalently K linearly generatesV ,) (c) pointed (K ∩ −K = ∅, so that it contains no nonnull subspace of V ), and (d)topologically closed (in the Euclidean metric topology, for finite dimension).

Such a positive cone induces a partial order ≥ on V , defined by x ≥K y := x − y ∈ K.(V,≥K), or sometimes (V,K), is called an ordered linear space. The Hermitian operatorson a finite-dimensional complex vector space, with the ordering induced by the coneof positive semidefinite operators, are an example. (A relation R is defined to be apartial order if it is reflexive (xRx), transitive (xRy & yRz ⇒ xRz) and antisymmetric((xRy & yRx) ⇒ x = y.) The partial orders induced by cones have the property that theyare “affine-compatible”: inequalities can be added, and multiplied by positive scalars.If one removes the requirement that the cones be generating, cones are in one-to-onecorrespondence with affine-compatible partial orderings. In fact, the categories of realvector spaces with distinguished cones, and partially ordered linear spaces, are equivalent.

We pause to motivate some of the seemingly technical conditions of regularity. A regularcone may represent the set of unnormalized probability states of a system, or a set ofspecifications of expectation values of observables. The normalized states may be gener-ated by intersecting it with an affine plane not containing the origin. Convexity is fairlyclearly motivated by operational considerations, such as those in the definition of convexeffect algebra above, or in the desire to have a normalized state set given by intersectingthe cone with an affine hyperplane be convex. Topological closure is required so that thecone has extreme rays, and the convex sets we derive by, for instance, intersecting it withan affine hyperplane, will have extreme points if that intersection is compact; then theKrein-Milman theorem states that these extreme points convexly generate the set. (Anaffine hyperplane is just a translation of a subspace: for d = 3, a 2-d hyperplane is a planein the sense of high school geometry.) In “empirically motivated” settings such as ours,in which the metric on the vector space will be related, via probabilities, to distinguisha-bility of states or operations, limit points can be as indistinguishable as you want fromthings already in the cone, so closing a cone cannot have empirically observable effects,and may as well be done if it is mathematically convenient. In the presence of someof the other assumptions, pointedness ensures that the intersection with an affine planecan be compact. Its appearance in the representation theorem for convex effect algebras(presumably essentially because the convex sets one gets via states tend to be compact

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intersections of an affine “normalization” plane with such a cone) is one “operational”justification for pointedness. Pointedness also has a clear geometric interpretation: if thesubspace K ∩ −K is one-dimensional, instead of a “point” at zero the cone could havean “edge,” which is why nonpointed cones are often referred to as “wedges”; of coursedim(K ∩ −K) > 1 is also possible for a nonpointed cone. The property of being gener-ating is often appropriate because any non-generating cone generates a subspace, and wemay as well work there. When several cones are considered at once, this might no longerbe appropriate.

An initial interval in such a space is an interval [0, u] defined as the set of things betweenzero and u in the partial ordering ≥K , i.e. x ∈ V : 0 ≤K x ≤K u. It is generatingif it linearly generates V . It can be viewed as a convex effect algebra by letting ⊕ bevector addition restricted to [0, u] and the convex structure be the restriction of scalarmultiplication. The representation theorem says any convex effect algebra is isomorphic(as a convex effect algebra) to some such linear convex effect algebra (via an affine map). Infinite-dimensional quantum mechanics the vector space and cone are Hd and the positivesemidefinite cone, and the interval referred to in the representation theorem is [0, I].

In addition to the requirements for states on an effect algebra, states on a convex effect al-gebra must satisfy ω(λa) = λω(a). The set of all possible states on a convex effect algebramay be characterized via a version of Lemma 3.3 of Gudder, Pulmannova, Bugajski, and Beltrametti(1999), which describes it for linear effect algebras [0, u]. First, some definitions. The dualvector space V ∗ for real V is the space of linear functions (“functionals”) from V to R;the dual cone K∗ (it is a cone in V ∗) is the set of linear functionals which are nonnegativeon K. Then Ω([0, u]), the set of all states on [0, u] when the latter is viewed as a convexeffect algebra, is precisely the restriction to [0, u] of the set of linear functionals f positiveon K and with f(u) = 1 (“normalized” linear functionals). The restriction map is abijection. Viewing things geometrically, the states (restricted functionals) are in one-to-one correspondence with the (unrestricted) functionals in the intersection of K∗ with theaffine plane in V ∗ given by f(u) = 1. Since any linear functional on the d2-dimensionalvector space Hd of Hermitian operators on Cd has the form X 7→ tr AX for some A,while the dual to the positive semidefinite cone in Hd is the set of such functionals forwhich A ≥ 0 (i.e., the positive semidefinite cone is self-dual (K = K∗)) this Lemma tellsus that the states of a finite-dimensional Hilbert space effect algebra are precisely thoseobtainable by tracing with density matrices ρ; in other words, the Gleason-type theoremfor POVMs is a case of this general characterization of states on convex effect algebras.This illustrates the power and appropriateness of this approach (and probably other con-vex approaches, in which similar characterizations probably exist) to empirical theories,and to problems in quantum foundations. Gleason’s theorem itself cannot be establishedin this way, because the effect algebra of projectors is not convex. However, there maybe a natural notion of “convexification” of effect algebras according to which [0, I] is theconvexification of the effect algebra of projectors. Interesting questions are then, whicheffect algebras can be convexified, and for which of those (as for the effect algebra of quan-tum projectors) convexification does not shrink the state-space. Conversely, we might askfor ways of identifying special subalgebras of effect algebras, composed of effects having

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special properties like “sharpness”, perhaps having additional structure such as that of anorthoalgebra, and investigate the relation between state-sets of effect algebras and thesesub-algebras.

5.3. Sequential operations

The operational approach I am advocating suggests that one consider what general kindsof “resources” are available for performing operations. Provided both system and ob-server are sufficiently “small” portions of the universe, it may be reasonable to supposethat the observer may use yet other subsystems (distinct from both observer and system)as an “apparatus” or “ancilla” to aid in the performance of these operations, that theapparatus may be initially independent of the system and observer, and that the com-bination of apparatus and system may be viewed as a system of the same general kindas the original system, subject to the same sort of empirical operational theory, with astructure, and a state, subject to certain consistency conditions with that of the originalsystem. (Convexity is a case of this, the ancilla functioning as “dice.”) It may be thatin some limits some of these assumptions break down, but it is still worth investigatingtheir consequences for several reasons: so that we can recognize breakdowns more easily,so that we may even acquire a theoretical understanding of when and why to expect suchbreakdowns, and because we may gain a better understanding of why empirical theoriesvalid in certain limits (say, small observer, small apparatus, small system) have the kindof structure they do.

Besides convex combination, other such elementary combinations and conditionings ofoperations should probably be allowed: essentially, the set of operations should be ex-tended to allow including them as subroutines in a classical randomized computation. (Ofcourse, this will not always be appropriate; for example, in investigating or constructingtheories that are not even classically computationally universal.) Among other things,this might get us the ⊕ operation previously obtained as the image of OR(∨) in Booleanpropositional logics about each operation’s outcomes, “for free,” as we can use classicalcircuitry to construct procedures whose outcomes naturally correspond to propositionalcombinations of the outcomes of other procedures, and will have the same probabilities asthose combinations. This leads us to the consider the possibility that the set of possibleoperations be closed under conditional composition. This means that given any operationM , and set of operations Mα, α ∈ M , there is an operation consisting of performingM , and, conditional on getting outcome α of M , then proceeding to perform Mα. Thisassumption is natural, but nevertheless substantive: one could imagine physical theoriesthat did not satisfy it. Some outcomes might destroy the system, or so alter it that wecan no longer perform on it all the procedures we could before. Nevertheless, it is worthinvestigating the structure of theories satisying the assumption (the theory of quantumoperations being one such case). The structures obtained when conditional composition isnot universally possible might turn out to be understandable as partial versions of thosewe obtain when it is always possible, or in some other way be easier to understand oncethe case of total conditional composability is understood. An operation in this frame-work, then, can be viewed as a tree with a single root node on top, each node of which

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is labelled by an operation and the branches below it labelled by the outcomes of theoperation, except that the leaves are unlabelled (or redundantly labelled by the labels ofthe branches above them). The interpretation is that the root node is the first operationperformed, and the labels of the daughters of a node indicate the operation to be per-formed conditional on having just obtained the outcome which labels the branch leadingto that daughter.

From now on, we mean by phenomenological theory a sequential phenomenological theory,i.e. one closed under conditional composition. If we extend a phenomenological theoryvia this requirement, the new outcome-set contains all finite strings of elements of the oldoutcome set. Given closure under conditional composition, a given string can now appearin more than one measurement. In order that the construction of dividing out operationalprobabilistic equivalence can work, we will have to require that the empirical probabilityof the string be noncontextual. We will also use a different notion of probabilistic equiva-lence: x ∼ y iff for any a, b, ω(axb) = ω(ayb), where x, y, a, b are outcome-strings. In ourcontext the noncontextuality assumption can actually be derived from the disjointnessof “elementary” operations (those not constructed via composition) and the assumptionthat the choice of operation at node n of the tree describing an operation constructedvia conditional composition cannot affect the probabilities of outcomes corresponding topaths through the tree not containing node n. This is how one might formalize a general-ization of the “no Everett phone” requirement suggested in Polchinski’s Polchinski (1991)article on Weinberg’s nonlinear quantum mechanics: the probability of an outcome se-quence cannot depend on what operation we would have done had some outcome in thissequence not occurred.

With suitable additional formalization of the notion of phenomenological operational the-ory, and appropriate definitions of ⊕ and a sequential product on the resulting equivalenceclasses, one can prove that dividing probabilistic equivalence out of such a set of empiricaloperations, in a manner similar to the construction of weak effect algebras via probabilisticequivalence, gives what I will call a weak operation algebra. The details will be presentedelsewhere. Here I will exhibit the quantum-mechanics of operations as a case of a generalstructure, an operation algebra (OA), which I view as the analogue, for operations, ofan effect algebra. The structure will be related to the notion of sequential effect algebra(SEA) studied by Gudder and Greechie (2000), but differ from it in important respects.It would be interesting to study when the set of effects of an OA forms a SEA.

Since this structure will be a partial abelian semigroup, with extra structure involvingonly the PAS operation ⊕, with a product meant to represent composition of operations,and additional axioms about how the two interact, we will discuss some more aspectsof PASes (following Wilce (1998)) before defining operation algebras. The reader mightwant to keep in mind the algebra of trace-nonincreasing completely positive maps (with⊕ as addition of maps and the product as composition of maps) as an example.

Recall that a PAS is a set with a strongly commutative and strongly associative partialbinary operation ⊕ defined on it. Define a zero of a PAS as an element 0 such that for

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any a, a ⊕ 0 = a. (Uniqueness follows.) If a PAS does not have a zero, it is trivial toadjoin one; we henceforth include its existence as part of a PAS. A PAS is cancellativeif x ⊕ y = x ⊕ z ⇒ y = z, positive if a ⊕ b = 0 ⇒ a, b = 0. The relation ≤ on a PASis defined by x ≤ y ⇔ ∃z x ⊕ z = y. Part of Lemma 1.2 of Wilce (1998) is that in acancellative, positive PAS ≤ is a partial ordering. In such a PAS, we define T as the setof top elements of the partial ordering (i.e. T = t ∈ O|a ⊕ t exists ⇒ a = 0). In acancellative PAS we define x ⊖ y as that unique (by cancellativity) z, if it exists, suchthat y ⊕ z = x. Define a chain in a partially ordered set P as a set C ⊆ P such that ≤restricted to C is total.

Definition 8 An operation algebra O is a cancellative, positive PAS equipped with a totalbinary operation, the sequential product, which we write multiplicatively. With respect tothe product, the structure is (OA5) a monoid (the product is associative) with (OA6) aunit 1 (semigroup is sometimes used as a synonym for this unital monoid structure). Theremaining axioms involve the interaction of this monoid structure with the PAS structure.(OA7) 0c = c0 = 0.(OA8) (a⊕ b)c = ab⊕ bc, a(b⊕ c) = ab⊕ ac (distributive laws).(OA9) 1 ∈ T .(OA10) Every chain in O has a sup in O.

Note that the sup mentioned in (OA10) is not necessarily in the chain. (OA10) says thatO is chain-complete; this is (nontrivally, and I am not certain whether choice or otherstrong axioms are required in the infinite case) equivalent to saying it is complete, meaningthat every directed subset of O has a sup in O. (A poset P is directed if for every subsetS of it, P contains an element x greater than or equal to everything in S.) The thinkingbehind (OA10) is that we are to conceive of the elements or “operations” in O as possibleoutcomes of procedures performed on a system, and each such outcome must be part ofat least one exhaustive set of such outcomes. Given how the ordering is defined, it mightseem natural therefore to require that all upward chains terminate; however, when thereare sufficiently many operations (and also, but not only, if continuous sets of outcomesfor a given operation are envisaged), as in the quantum case, it could be reasonable toallow (what is certainly possible in the quantum case) chains that do not terminate, buthave a limit point (the sup mentioned in (OA10)).

Our structure is not an effect algebra because we do not assume it is (as a PAS) unital(i.e., has at least one unit). A unit of a PAS is an element u such that for any a, there isat least one b such that a ⊕ b = u. In a cancellative, positive, unital PAS (equivalently,effect algebra) there is a unique unit (the sole element of the top-set T ). Axiom (OA10)might need strengthening in order to obtain some of the results one would like. Notably,we would like to have a representation theorem in which the operations belong to a conein a vector space (and thus belong to an algebra in one of the usual mathematical senses,of a vector space with an appropriate product). Aside from belonging to a cone, thespecial nature of the convex set of operations in such a representation theorem would beexpressed by an additional requirement, deriving from (OA10), which would specialize to

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the trace-nonincreasing requirement in the case of the quantum operation algebra (andgeneralize the initial interval requirement in the analogous representation theorem foreffect algebras).

We shall now show that quantum mechanics provides an example of this structure. Werefer to the set of linear operators on Cd as B(Cd).

Proposition 1 The set of trace-nonincreasing completely positive linear maps on B(Cd),with the identity map I as 1, the map M defined by M(X) = 0 for every X as 0, ordinaryaddition of maps as linear operators, restricted to the trace-non-increasing interval, as ⊕,and composition of maps as the sequential product, forms an operation algebra. Its top-setT is the set of trace-preserving maps.

Proof: The commutativity (OA2) and associativity (OA1) of ⊕ and the behaviorof 0 (OA7), and the unital monoid structure (OA5 and 6) are immediate. Cancellativityholds for addition in any linear space, so since ⊕ is here a restriction of addition ona linear space of linear maps, it is cancellative (OA3). It is positive (OA4) becauseA+B = 0 ⇒ A,B = 0 for A,B in a pointed cone (such as the cone of completely positivelinear maps). (OA8) follows from the distributivity of multiplication of linear operatorsover addition of linear operators. The top-set T is the set of trace-preserving operations,which follows from the easy observation that if you add any operation besides the zerooperation to a trace-preserving operation, the result is not trace-nonincreasing. (OA9)follows since the identity operation is trace preserving. (OA10) involves an elementarytopological argument which will be omitted here.

We note the interpretation of ⊕ and ⊖ in terms of the HK representation of a map A interms of operators Ai (operators such that the map acts as X 7→

∑iAiXA

†i). Modulo

irrelevant details of indexing, the HK representation sequence Ai is a multiset [A] ofoperators A such that A†A ≤ 1. A ⊖ B exists if there are HK representations [A], [B]such that [B] is a submultiset of [A]. (Equivalently, there are standard HK representationsequences Ai and Bi such that Bi is an initial segment of Ai, i.e. B(X) =

∑iAiXA

†i

where i ranges over the first k Ai.) Thus it is obvious that A⊖ B will not always exist.

We define a weak operation algebra to satisfy all the above axioms except that associativityis replaced with weak associativity (whose statement is the same as in the definition ofweak effect algebra). With suitable additional formalization of the notion of sequentialphenomenological theory and sequential probabilistic equivalence, and definitions of ⊕and sequential product on the equivalence classes, one can show:

Theorem 2 The set of equivalence classes obtained by dividing the notion of operationalprobabilistic equivalence defined above out of a phenomenological operational theory, hasa natural weak operation algebra structure.

Note that if we have operational limits on conditional composition, as discussed above, wemight accomodate that by modifying the notion of operation algebra (or WOA) to make

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the multiplicative monoid structure partial. It would then be interesting to investigatethe conditions under which this partial structure is extendible to a total one (as well asthe conditions under which a WOA can be completed to an OA).

We can add a convex structure to an OA with little difficulty. We just introduce a map ofmultiplication by scalars in [0, 1] (i.e. a map from [0, 1]× O → O) such that the axioms(C1–C4) of convex effect algebras hold, and also (αa)b = α(ab) = a(αb) (COA15). Weexpect such a structure to again emerge from an operational equivalence argument appliedto a suitable notion of convex operational phenomenological theory.

6. Dynamics and the combination of subsystems in operational theories

The operation algebra approach sketched above implicitly includes a kind of dynamics,although without explicit introduction of a real parameter for time. Probably some opera-tion algebras are extendible to have a notion of time. However, in the quantum operationalgebra given above the assumption is that any completely positive evolution can beachieved. The time taken is neglected, and the temporal element of the interpretation isonly the primitive one that when one measurement is done conditional on the result ofanother, it is thought of as done after the result of the first is obtained. A more substan-tial notion of time might be introduced in many different ways by adding structure to theoperation algebra, e.g. by some consistent specification of how long each evolution takes,or by the assumption that each evolution can be done in any desired finite amount oftime. The latter is a very strong assumption. In some cases, one might have a continuoussemigroup structure related (with scheduling constraints) to their sequential product. Arealistic consideration of these matters would involve a much more detailed account ofthe interactions between apparatus and system that are actually available. This is animportant part of the project I propose, but I will not pursue it much here. It remindsus, though, of one of the important lessons of QIP for foundations mentioned in Section2: that which operations are possible may depend on the resources available, and thatthe beautiful structures one sometimes encounters as operational theories may be ideal-ized. In particular, much of the attempt to implement QIP involves struggling with thelimitations imposed by the limited nature of the subsystems, and interactions, physicsmakes available. It is important to incorporate such limitations in operational structures.Barnum et al. (2002) is one approach to this, with the resources available for control andobservation limited (for example) to those definable via a Lie subalgebra of the full Liealgebra sl(d) appropriate to arbitary quantum operations. Physics includes much morethan just Hilbert space: preferred bases or tensor product structures, symmetries, thewhole business of representation theory. Another approach to involving this “more” inoperational theories has been the inauguration, particularly in works such as Foulis (2000)and Wilce (2000), of a theory of group actions on empirical quantum logics.

An important part of the project of combining operational empirical logic and QIP ideasto investigate whether or not physics can provide an overarching structure unifying per-spectives is to understand the operations available in an operational theory in terms of

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interactions with apparatus and/or environment. In particular, if we have a way, such asthe tensor product in quantum mechanics, of describing the combination of apparatus Aand system S as subsystems of a larger system L, we will probably want to require that theevolution induced on S by doing an operation on the larger system is, under appropriatecircumstances, one of the operations our theory describes as performable on the smallersystem. “Appropriate circumstances” probably means that the apparatus should be ini-tially independent of the system, which in turn requires that the notion of combination ofsubsystems have a way of implementing that requirement. Such assumptions bear closescrutiny, though, as they may be just the sort of thing that becomes impossible in certainlimits. Some, such as (Ford et al., 2001), have argued for the physical relevance of somesituations in which open systems are analyzed without the initial independence assump-tion. Independence works well in the case of completely positive quantum operations,though: indeed, all such operations can be implemented via a reversible interaction withapparatus. Consideration of categories, such as convex operation algebras and generaliza-tions of these, that describe dynamics is probably the most promising way to investigatesuch questions. Possibly the category-theoretic notion of tensor product will be definedfor these categories. One could then examine, for example, whether the tensor productof two Hilbert-space CP-operation algebras is the operation algebra of CP-maps on thetensor product of the Hilbert spaces. I doubt that it is.

To define the category-theoretic tensor product requires the notion of bimorphism. Forcategories whose objects are sets with additional structure, and whose morphisms arestructure-preserving mappings we can define a bimorphism of A,B as function φ : A×B →T , where T is another object in the category, and φ has the property that for everya ∈ A, φa : B → T defined via φa(b) = φ(a, b) is a morphism, and similarly with the rolesof A,B reversed. In the category of vector spaces, for example, it is just a bilinear map.

Definition 9 The tensor product A⊗B is a pair (T, τ), where T is another object in thecategory (also often called the tensor product) and τ : A× B → T is a bimorphism, andany bimorphism from A×B factors through T in a unique way, and T is minimal amongobjects for which such a τ exists.

To say τ factors through T in a unique way is just to say that for any bimorphismβ : A⊗ B → V , there is a unique φ : T → V such that β is τ followed by φ. Minimalityin a set means not a subobject of any object in the set. Probably the uniqueness of thefactorization is therefore redundant.

There is an “operational” motivation of this construction when it is applied to categorieslike effect algebras, operation algebras, etc...: it implements the notion that the twostructures being combined appear as potentially “independent” subsystems of the largersystem, in a fairly strong sense that one can do any operation (or get any outcome) onone subsystem while still having available the full panoply of operations (outcomes) onthe other.

The category-theoretic tensor product of ordered linear spaces (vector spaces with dis-

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tinguished regular cones) is not well defined: more structure is needed. More precisely,while various constructions having the universal property (all bimorphisms factor throughthem) can be made, there is not a unique minimal one.

For a variety of operational structures one might use to describe quantum mechanicalstatics, including test spaces, orthoalgebras, and effect algebras, the tensor product is notthe corresponding operational structure for the tensor product of Hilbert spaces. Thiscould indicate that the structure describing statics requires more specialized axioms, stillconsistent with quantum mechanics, and then the tensor product in this new category,call it Z, will come out right in the Hilbert space case. It could also be that the difficultyis the static nature of the categories. Indeed, the category-theoretic tensor product oftest spaces or effect algebras includes measurements whose performance would seem toinvolve dynamical aspects. These are measurements describable as the performance ofa measurement M on system A, followed by the performance of a measurement Mα,on B, where which measurement Mα is performed is conditional on the outcome α ofthe A-measurement. The the tensor product of effect algebras must contain all productoutcomes, and it can be characterized as the effect algebra “generated” by requiringthat it contain all the “1-LOCC” (local operations with one round, in either direction,of classical communication) measurements just described. Fuchs’ (2001a) “Gleason-liketheorem for product measurements” effectively does this construction for the case ofHilbert effect algebras. It is fairly elementary to show that the tensor product of EA’scan also be characterized as the minimal “influence-free” effect algebra containing allproduct measurements (i.e. in which we can do all pairs of measurements one on A, oneon B, with no communication). Freedom from influence of B on Ameans that for all stateson the object, the probabilities of the outcomes of an A measurement, performed togetherwith an independent B measurement, cannot be affected by the choice of measurementon B. Influence freedom means freedom from influence in both directions. Both of thesecharacterizations provide strong operational motivation for the category-theoretic tensorproduct in this situation. Each is easily established starting from the other, and a similarconstruction of a “directed” product, in which 1-LOCC operations are allowed in onedirection only, rules out “influence” in the direction opposite the communication. Thesethings are also true, and were in fact first established for, test spaces (Foulis and Randall,1981) and orthoalgebras Bennett and Foulis (1993).

The difficulty, in the quantum case, is that the tensor product of orthoalgebras or ef-fect algebras, while it must contain measurements of effects that are tensor products ofAlice and Bob effects, and, through addition of effects, all separable effects, does notcontain “entangled” Alice-Bob effects. The separable effects span the same vector spaceB(Cd⊗Cd) ∼= Hd2 of d

2×d2 Hermitian matrices (where A,B both have dimension d) as thefull set of effects on Cd⊗Cd, but they are the interval [0, I] in the separable cone, not theinterval [0, I] in the positive semidefinite cone. Consequently the available states, whilethey must be linear functionals of the form A 7→ tr AX for d2 × d2 Hermitian X , are thenormalized members of the separable cone’s dual, rather than of the positive semidefinitecone’s dual, so X in the functional A 7→ tr AX is not necessarily positive semidefinite.The separable cone being properly contained in the positive semidefinite one, its dual

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properly contains the positive semidefinite one’s dual, so that not only are we restrictedto fewer possible measurements, but their statistics—even those of independent A,Bmeasurements—can be different from the quantum ones (although all quantum states arealso possible states). Stated in more quantum information-theoretic terms: some non-positive operators X are nonpositive in ways that only show up as negative probabilitiesor nonadditivity when we consider entangled measurements: since in the effect-algebraor orthoalgebra tensor product we don’t have entangled measurements available to “di-rectly detect” this nonpositivity, these are admissible states on these tensor products.Indeed, as observed in Wilce (1992), they are isomorphic to the Choi matrices (block ma-trices whose blocks Mi,j are T (|i〉〈j|)) of positive, but not necessarily completely positive,maps T (although the normalization condition (trace-preservation) appropriate for suchmaps is different from the (unit trace) normalization condition appropriate for states).Of course, the nonpositivity of the operator can be “indirectly detected” by tomographyusing separable effects, since these effects span the space of Hermitian operators.

One obvious solution to the problem would be to introduce axioms that would prohibitthis divergence between the existence of entangled states and nonexistence of entangledmeasurements. Mathematically, this divergence reflects the important fact that the pos-itive semidefinite versus separable effect algebras on Cd ⊗ Cd are differentiated by theproperties of the corresponding cones: the former, but not the latter, being self-dual.Self-duality is a natural and powerful mathematical requirement on cones, but a verystrong, and arguably not operationally motivated, one. Self-duality is an important partof the essence of quantum mechanics, so we should strive hard to understand its oper-ational motivation and implications. The cones for classical effect algebras can also beself-dual: e.g. the algebra of fuzzy sets of d objects. An axiom related to self-duality, vio-lated by the tensor product of Hilbert effect algebras, is the “purity is testability axiom.”We develop some concepts before formulating it.

Definition 10 An effect-algebra theory is a pair 〈E ,Υ〉 where E is an effect algebra, Υa convex set of states on that effect algebra. An effect t passes a state ω if ω(t) = 1. Aneffect t is a test for ω in theory 〈E , υ〉 if t passes ω ∈ Υ and for no state σ 6= ω, σ ∈ Υ,does t pass σ. A state ω ∈ Ω is testable in 〈E ,Ω〉 if a test for it exists in E .

Note that Υ may be smaller than Ω(E), the set of all possible states on E . We now assumeeffect algebras are convex. If two tests pass ω, so does any mixture of those tests. Lett be a test for ω, then for σ 6= ω, (λω + (1 − λ)σ)(t) = λω(t) + (1 − λ)σ(t) < 1, i.e. tcannot test any mixture of ω with something else. Although a test thus tests a uniquestate, it is not necessarily the case that a testable state has a unique test. Let t test ω;suppose ω = λσ + (1 − λ)τ . Then 1 = ω(t) = λσ(t) + (1 − λ)τ(t). This implies thatσ(t) = τ(t) = 1, hence by the fact that t tests ω, σ = τ = ω. In other words, only pure(extremal) states can be testable. We will be interested in Axiom 1: all pure states aretestable. To study the consequences of this axiom, we introduce a basic notion in convexsets.

Definition 11 A face of a convex set C is an F ⊆ C such that for every point p ∈ F ,

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all points in terms of which p can be written as a convex combination are also in F . Inother words, for λi ≥ 0,

∑i λi = 1,

∑i λixi ∈ F ⇒ (∀i, xi ∈ F ) .

Thus a face of C is the intersection of the affine plane it generates with C. The set offaces, ordered by set inclusion, forms a lattice. This lattice characterizes the convex set.(up to affine isomorphism, which is the proper notion of isomorphism for convex sets sinceaffine transformations y 7→ Ay + b commute with convex combination).

Proposition 2 The theory 〈E(Cd)⊗E(Cd),Υ〉 violates Axiom 1 unless Υ is contained inthe set of separable states. In particular, 〈E(Cd)⊗ E(Cd),Ω(E(Cd)⊗ E(Cd))〉 violates it.

Proof: The proof proceeds by showing that the only states testable in E(Cd)⊗E(Cd)are pure product states. Then if Axiom 1 is satisfied, the extremal states of Υ areproduct states, so Υ is a face of the convex set of separable states. Let tr X = 1 and〈χ|〈ψ|X|ψ〉|χ〉 ≥ 0 for all product states |φ〉|χ〉, so that A 7→ tr AX is a state. Testabilitymeans there is a separable A with trace between zero and one (separable effect) such that:1 = tr AX . The first requirement on A says that A =

∑i λi|χi〉|ψi〉〈ψi|〈χi| (for λi >

0,∑

i λi ≤ 1, |χi〉, |ψi〉 normalized). Thus tr AX = 1 becomes∑

i λi〈χi|〈ψi|X|ψi〉|χi〉 = 1,which can only hold if one of the λi = 1, and for that i, 〈χ|〈ψi|X|ψi〉|χi〉 = 1. Then(dropping the subscript) X = |χ〉|φ〉〈φ|〈χ|+Xπ,⊥ +X⊥,π +X⊥,⊥ . This is a resolution ofX into components in four subspaces of the space of operators on Cd⊗Cd: the space π, πof operators on the one-dimensional Hilbert space π spanned by the pure product state,the space π,⊥ of operators taking π to π⊥, the space ⊥, π going the other way, and thespace ⊥,⊥ of operators on π⊥. The middle two pieces are manifestly traceless, so thelast one must be traceless for tr X = 1 to hold. However, tr X⊥,⊥ =

∑ij〈i|〈j|X|j〉|i〉 in

a product basis |i〉|j〉 for ⊥. Each 〈i|〈j|X|j〉|i〉 must be positive since tr X⊥,⊥A = tr XAfor A ∈⊥,⊥. So for X⊥,⊥ to be traceless, they must all be zero, and X = |χ〉|φ〉〈φ|〈χ|plus possibly some traceless stuff which does not affect the induced state.

Note that we can have a theory on E(Cd)⊗ E(Cd) satisfying the axiom of testability, butonly if the state space is contained in the dual of the cone generated by the effect algebra.This suggests that the axiom, if required of the full state space Ω of an effect algebra, ispushing us towards the idea that the cone be self-dual.

Testability is very natural, and has a long history in quantum logic (e.g. Mielnik (1969)and probably Ludwig (1983a; 1985)). Theories which are the full state spaces of lineareffect algebras that are initial intervals in self-dual cones satisfy it. This axiom makescontact with the “property lattice” quantum logics of Jauch (1968) and Piron (1976).(See Valckenborgh (2000, pp. 220–221)). It is also related to Ruttiman’s Ruttiman(1981) notion of “detectable property.” Jauch and Piron’s notion of property roughlycorresponds to effects (or the analogues in other quantum structures, since most of theirwork was done before effect algebras were formalized in the quantum logic community) ewhich can have probability one in (“pass”) some states. Those states are said to “possessthe property [e]”. Properties are equivalence classes of effects that pass the same set of

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states. They construct a lattice of properties for an empirical theory (set of states onsome quantum structure).

Axiom 1 relates the lattice of faces of a convex set of states on an effect algebra to theproperty lattice of that theory. The extremal states are minimal in the face lattice; theaxiom says there are “minimal properties” possessed by those states: minimal in thesense that no other state posesses them. I am not certain if this is minimality in thesense of Piron’s property lattice, but it seems likely (perhaps under mild conditions).A generalization of Axiom 1 asserts, for each face of the state-set, the existence of a“property” of being in that that face (an effect passing the states of that face and noothers). A similar axiom of Araki (1980) concerns “filters” for higher dimensional faces,but this also involves “projection postulate-like” dynamics associated with the filtering.Araki also uses, as an assumption, the symmetry or “reciprocity” rule, satisfied in thequantum-mechanical case, that can be formulated once a correspondence χ↔ eχ betweenextreme states χ and tests eχ for them has been set up. Reciprocity requires that χ(eφ) =φ(eχ) . It is not clear to me whether the extreme states → effects correspondence must beone-to-one instead of many-to-one in order to be able to formulate the axiom, or whetherone-to-oneness might be a consequence of it. (Faces play an important role in Ludwig’swork as well, as do statements reminiscent of Axiom 1, so Ludwig’s argument may turnout to be similar.)

Araki credits Haag for emphasizing to him the importance of the reciprocity axiom. In thesecond edition of his book, Haag (1996) includes a informal discussion of the foundationsof quantum mechanics based on the convex cones framework. He, too, uses Axiom 1, anda generalization associating faces of the state space (one-to-one!) with “propositions.”These “propositions” are effects passing precisely the states of the face, and minimalamong such effects in the sense of a probabilistic ordering of effects e1 ≤ e2 := ∀ω ∈Υ ω(e1) ≤ ω(e2) . This is a different strategy from the Jauch-Piron equivalence class onefor getting uniqueness of the effect associated to a face, but it is closely related to it.Jauch and Piron were trying to get by with less reference to probabilities. Haag also usesthe reciprocity axiom, which he argues imposes self-duality.2

Haag also gives some operational motivation for an additional assumption, that of homo-geneity of the cone. This says that the automorphism group of the cone acts transitivelyon its interior. (For any pair x, y of interior points, there is an automorphism taking x toy.) Interpret cone automorphisms as conditional dynamics; then homogeneity, at least forself-dual cones, means that any state is reachable from any other by dynamics conditionalon some measurement outcome. This is not self-evident but seems natural. If you can’tprepare any state starting from any other state, with a nonzero probability of success,the state space might “fall apart” into pieces not reachable one from the other (orbits ofthe automorphism group). Or while some pieces might still be reachable from all others,

2Haag uses uses the notion of self-polarity, but for our type of cone, this is the same as self-duality. Thepolar of a convex body C is the set of linear functionals L such that L(x) ≤ 1 for all x ∈ C; the polar ofa cone is the negative of the dual cone, since whenever L(x) is positive, L(x′) is greater than 1 for x′ alarge enough positive multiple of x. Since the negative of a cone is isomorphic to that cone, a self-polarcone is self-dual.

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going the other way might not be possible: there would be intrinsically irreversible dy-namics, even conditionally. A more detailed study of operational theories whose effectsare naturally represented in a non-homogeneous cone, or whose state-space generates one,would be desirable (either with or without self-duality). The “falling apart” into orbits ofthe automorphism group may be acceptable in a theory of a perspective involving radicallimitations on our ability to prepare states: going from one orbit to another might re-quire a more powerful agent than the one whose perspective is being considered, but theconsequences of such an agent’s actions might be observable by the less powerful agent.Entanglement is such a situation: the perspective of the set of local agents, even withthe power to communicate classically, allows for pairs of states with different statistics forobservables implementable by local actions and classical communication (LOCC), suchthat it is impossible, even conditional on a measurement outcome, to prepare one startingfrom the other via LOCC W. Dur (2000). The LOCC perspective of the local agents isnot usually taken as a “subsystem” in quantum mechanics, so these sorts of perspectivescan there be taken as derivative rather than fundamental; but perhaps in other situationsnonhomogeneous perspectives could be more fundamental.

In finite dimensions, as Haag points out, homogeneous self-polar cones are known (e.g.(Vinberg, 1965)) to be isomorphic to direct products of the cones whose faces are thesubspaces of complex, quaternionic, or real Hilbert spaces. (Extensions of these resultsto infinite dimensions are obtained in Connes (1974).) The factors in the direct productcan be thought of as “superselection sectors;” classical theory would be recovered whenthe superselection sectors are all one-dimensional (at least in the complex and real cases).Araki (1980) obtains a similar theorem except the effects get represented as elements ofa finite dimensional Jordan algebra factor. These are isomorphic to to n × n Hermitianmatrices over R,C, or the quaternions H, or a couple of exceptional cases (spin factorsand 3 × 3 Hermitian matrices over the Cayley numbers). He also gives arguments forpicking the complex case, based on the properties of composition of subsystems in thevarious cases. Araki’s argument is that “independence” of the subsystems should beexpressed by dim V = ( dim V1)( dim V2) for the algebras. But, “essentially becausethe tensor product of two skew-Hermitian operators is Hermitian”, we have dim V >( dim V1)( dim V2) except in trivial cases, when we take the V ’s to be the algebras ofHermitian matrices over real Hilbert spaces H1, H2, and their tensor product. (A relatedrequirement plays a similar role in Hardy (2001a,b).) ForQ there is not even a quaternion-linear tensor product. The bottom line is that “the complex field has the most pleasantfeature that the linear span of the state space of the combined system is a tensor productof [the state spaces of the] individual ones.” There are probably important operationaland information-theoretic distinctions between the cases which merit closer study. In thereal case, the key point is that in contrast to the complex case, states on the “natural”real composite system are not determined by the expectation values of local observables.

Like homogeneity, self-duality and reciprocity may be related to the coordination of per-spectives into an overall structure. In a “spin-network” type of theory, the edges of agraph are labelled with representations of a Lie or quantum group (su(2), for spin net-works), which are Hilbert spaces. The vertices are associated to “intertwiners” between

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those representations. A state might be associated with, say, a partition of the graph bya hypersurface cutting it into two parts, “observer” and “observed.” If the hypersurfacehas two disconnected parts, the associated Hilbert space will be the tensor product of theones associated with the parts; otherwise, the representation is made out of the represen-tations labelling the cut edges, in a way determined by the intertwinings at the verticesbetween them. One has the same Hilbert space whichever piece one takes as “observer”vs. “observed.” However, it is likely that the role-reversal between observer and observedcorresponds to dualization, and the result that both correspond to the same Hilbert spacewill only hold in theories in which the structure describing a given perspective—here,the Hilbert space associated with the surface—is self-dual. To attempt to actually showsomething like this would involve a project of trying to construct “relational” theorieslike the Crane-Rovelli-Smolin theories, but with other empirical theories playing the roleof Hilbert spaces and algebras of observables on them. A simple first example might be“topological classical field theories,” if these can consistently be defined. In these general“pluralistic structures” coordinating perspectives, one might hope to find a role for self-duality and the reciprocity axiom, and perhaps homogeneity as well. For the differentempirical structures associated with different surfaces to relate to each other in a “nice”way, it might be necessary that the structures be defined on self-dual cones or exhibitreciprocity. Another suggestion that bears more detailed investigation, perhaps also inthe “relational” context since there time is sometimes taken as emergent, is due to Haag,who says, “[reciprocity] expresses a symmetry between “state preparing instruments” and“analyzing instruments” and is thus related to time-reversal invariance.”

7. Tasks and axioms: toward the marriage of quantum information scienceand operational quantum logic

QIP emphasizes how the conceptual peculiarities of quantum mechanics allow us to per-form tasks not classically possible. This suggests we these formulate tasks, or the as-sociated concepts, in ways general enough to try to characterize different operationaltheories by whether or not these tasks can be performed in them, or by the presence orabsence of conceptual phenomena such as: superposition, complementarity, entanglement,information-disturbance tradeoffs, restrictions on cloning or broadcasting, nonuniquenessof the expression of states as convex combinations of extremal quantum states (versusthe uniqueness classically), and so forth. An outstanding example involves cryptographictasks (Fuchs, 2001a; Clifton et al., 2002). But even before the upsurge of interest in quan-tum information science, conceptual peculiarities like superposition (Bennett and Foulis,1990) and nonunique extremal decomposition (Beltrametti and Bugajski, 1993) were be-ing generalized and studied in empirical/operational quantum logic.

Assumptions and tasks involving computation should also be investigated; In particular,it would be interesting to establish linkages between complementarity, or superposition,and computational speedup in a general setting. Or some conjunction of properties,such as no instantaneous communication between subsystems, common to quantum andclassical mechanics, might be seen to imply no exponential speedup of brute-force search

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in a general setting. I claimed above that key aspects of using an operational pointof view in foundational questions were understanding notions of subystems and systemcombination, and understanding dynamics. For information-processing or computation,both of these issues are of the utmost importance. Since the environment which inducesnoise in a system or the apparatus used by an information-processing agent must beconsidered together with the system, a notion of composite system is needed. And notionsof composition of systems or of dynamics are basic to computational complexity, where thequestion may be how many bits or qubits are needed, as a function of the size of an instanceof a problem (number of bits needed to write down an integer to be factored, say) to solvethat instance. The very notion of Turing computability is based on a factorization of thecomputer’s state space (as a Cartesian product of bits, or of some higher-arity systems),in terms of which a “locality” constraint can be imposed. The constraint is, roughly,that only a few of these subsystems can interact in one “time-step.” The analogousquantum constraint allows only a few qubits to interact at a time. In general operationalmodels, some notion of composition of systems, such as a tensor product, together witha theory describing what dynamics can be implemented on a subsystem, could allowfor generalized circuit or Turing-machine models. Another way of obtaining a notion ofresources is to specify a set of dynamical evolutions to which we ascribe unit cost, and a setof measurements viewed as computationally easy. More generally, we might specify a costfunction on evolutions and measurements. A formal treatment will require us to say howwe interface the given operational model with “classical” computation. We could specifya set of measurements-with-conditional-dynamics (“instruments”) viewed as taking unitcomputational time, and allow the conditioning of further dynamics and measurementon the results of the measurement in question. Subtleties could arise in counting thecomputational cost of the classical manipulations required by such conditioning. Countingone elementary operation in some chosen classical computational model as costing thesame as one in the general operational model is one reasonable approach (at least if thegeneral model can simulate classical computation polynomially). More simply, performthe algorithm in the general operational setting by evolution without explicit measurementand classical control, and specify a “standard” measurement to be performed at the end(and a standard procedure for mapping the measurement result to the set of possiblevalues of the function being computed). In non-query models, it is important that notjust any measurement be allowed at the end, since if the dynamics consists of all effect-algebra endomorphisms, say, any computation can be done by making one measurement.

8. Conclusion

In this paper, I have promoted a particular project for harnessing the concepts of quantuminformation science to the task of illuminating quantum foundations. This project is togeneralize tasks and concepts of information science beyond the classical and the quan-tum, to abstract and mathematically natural frameworks that have been developed forrepresenting empirical theories; and to use these tasks and concepts to develop axioms forsuch theories, having intuitively graspable, perhaps even practical, meaning, or to develop

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a better understanding for the operational meaning of existing axioms. The main originaltechnical contributions are Theorem 1 showing that any phenomenological theory natu-rally gives rise to a weak effect algebra, which is essentially the image of the propositionallogic of statements about measurement outcomes under identification of probabilisticallyequivalent outcomes, and the introduction of the notions of operation algebra and weakoperation algebra. These results and concepts are likely closely related to other work inoperational quantum logic and the convex approach; I think they provide an appropriateframework for the project.

Within the scope of this project, I have emphasized what I think will be key aspects:

• A “perspectival, operational” approach to describing empirical theories, taking theprobabilities of outcomes of operations an agent may do on the system as primary,and stressing that the structure of an empirical theory depends on the agent doingthe operations as well as on the subystem the operations are done on.

• The structures of effect algebras and weak effect algebras, test spaces, and proposi-tion lattices for observations, as well frameworks of “operation algebras” and “weakoperation algebras” introduced here to encompass both dynamics and observables.

• A justification of weak effect and operation algebras through relations of “proba-bilistic equivalence,” and “sequential probabilistic equivalence,” as natural represen-tations of very general classes of phenomenological theories. Gleason-type theoremstake on a fresh aspect from this point of view.

• Convexity, and the resulting representations it makes possible in ordered linearspaces (real vector spaces with distinguished regular cones), and various mathemat-ically natural axioms it suggests, such as homogeneity and self-duality.

• The significance of other natural operational desiderata, such as the idea that any-thing implementable via interaction with an independent ancilla should be consid-ered an operation, or the idea that “evolve and then measure” should be considereda kind of measurement.

• The importance of attempts, like the Rovelli-Smolin “relational quantum mechan-ics,” topological quantum field theories, spin networks, and “spacetime foams,” tointegrate agents’ perspectives into a coherent whole, as special relativity does withits reference frames. The use of “integrability of perspectives into a coherent whole,”as a possible source of axioms about the nature of perspectives (self-duality or ho-mogeneity of the cones used to represent them?), how they combine (via tensorproducts or some other rule?), and so forth.

Acknowledgments

Discussions over the years with Carlton Caves, Dave Foulis, Chris Fuchs, Leonid Gurvits,Lucien Hardy, Richard Jozsa, Manny Knill, Eric Rains, Rudiger Schack, Ben Schumacher,

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and Alex Wilce, among others, have influenced my thoughts on these matters.

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